CANBERRA, Australia—Police carried out early morning raids in two major Australian cities, including Sydney, in a strike aimed at disrupting what they said were plans by local Islamic State supporters to behead members of the public.

Almost 900 state and national police, as well as domestic-spy-agency officers, were involved in the counterterrorism raids, to prevent what Prime Minister Tony Abbott dubbed "demonstration killings," a reference to their intended impact of showing the reach of Islamic State beyond the...